A T-shirt manufacturer is planning to expand its workforce. It estimates that the number of T-shirts produced by hiring $x$ new workers is given by $T(x)=-0.75 x^{4}+24 x^{3}, 0 \leq x \leq 24$. When is the rate of change of $T$-shirt production increasing and when is it decreasing? ${ }^{\circ}$ What is the point of diminishing returns and the maximum rate of change for T-shirt production? Graph $T$ and $T^{\prime}$ on the same coordinate system.

Step 1 ：First, we need to find the derivative of the function $T(x)$, which represents the rate of change of T-shirt production. The derivative of $T(x)$ is given by $T'(x)=-3x^{3}+72x^{2}$.

Step 2 ：Next, we need to find the critical points of $T'(x)$, which are the points where the derivative is either 0 or undefined. To find these points, we set $T'(x)$ equal to 0 and solve for $x$. This gives us the equation $-3x^{3}+72x^{2}=0$.

Step 3 ：We can factor out $-3x^{2}$ from the equation to get $-3x^{2}(x-24)=0$. Setting each factor equal to 0 gives us $x=0$ and $x=24$ as the critical points.

Step 4 ：To determine where the rate of change is increasing or decreasing, we can use the second derivative test. The second derivative of $T(x)$ is $T''(x)=-9x^{2}+144x$.

Step 5 ：We can then plug in the critical points into $T''(x)$ to determine the concavity at these points. If $T''(x)>0$, then the function is concave up and the rate of change is increasing. If $T''(x)<0$, then the function is concave down and the rate of change is decreasing.

Step 6 ：Plugging in $x=0$ into $T''(x)$ gives us $T''(0)=0$, which means the rate of change is neither increasing nor decreasing at $x=0$.

Step 7 ：Plugging in $x=24$ into $T''(x)$ gives us $T''(24)=-9(24)^{2}+144(24)=-5184$, which is less than 0. Therefore, the rate of change is decreasing at $x=24$.

Step 8 ：Since the rate of change is decreasing at $x=24$, the point of diminishing returns is at $x=24$.

Step 9 ：To find the maximum rate of change, we need to find the maximum value of $T'(x)$. Since $T'(x)$ is a cubic function, it has no maximum value. Therefore, the maximum rate of change is infinite.

Step 10 ：Finally, we can graph $T(x)$ and $T'(x)$ on the same coordinate system to visualize the rate of change of T-shirt production.